The idea

Cumulative binomial probabilities

Finding P(X ≤ r), P(X < r), P(X ≥ r), P(X > r) and P(a ≤ X ≤ b) for a binomial distribution by building everything from the cumulative function P(X ≤ r) (binomcdf), with care over the off-by-one when an inequality is strict or "at least". Includes the complement for "≥" and "two-stage" problems where the success of one binomial becomes the trial of another.

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Statistics · Statistical distributions

Cumulative binomial probabilities

Why it works

Build everything from P(X ≤ r)

X \sim B(15, 0.48)

— find

P(X \ge 5)

. Done term by term that is eleven binomial calculations,

P(5) + P(6) + \dots + P(15)

, each one a formula in its own right. The calculator's cumulative function collapses the whole job into one line —

P(X \le r) = P(X=0) + P(X=1) + \dots + P(X=r),

given directly as

\text{binomcdf}(n, p, r)

— and every range an exam can ask for can be rebuilt from it. The entire skill of this lesson is translating the inequality correctly:

\begin{array}{l|l} \text{You want} & \text{Compute it as} \\ \hline P(X \le r) & \text{binomcdf}(n, p, r) \\ P(X < r) & P(X \le r-1) \\ P(X \ge r) & 1 - P(X \le r-1) \\ P(X > r) & 1 - P(X \le r) \\ P(a \le X \le b) & P(X \le b) - P(X \le a-1) \end{array}

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